On Extremal Integers of Prescribed 2-Adic Valuation Under Bounded Integer Complexity: A Strengthening of the Powers-of-Two Conjecture

Let ||n|| denote the integer complexity of n: the minimum number of ones required to represent n using ones, addition, and multiplication. The central open problem in integer complexity, posed by Selfridge and designated the "Question of Questions" by Iraids et al., asks whether ||2^m|| = 2m for every m >= 1. We formulate a strictly stronger conjecture: for every positive integer n, the bound ||n|| >= 2v_2(n) holds, where v_2(n) is the 2-adic valuation of n. We prove that this bound, together with the known extremal structure of odd integers, yields a complete characterization of the largest integer of prescribed 2-adic valuation achievable within a given complexity budget: M(K,a) = 2^a O(K - 2a), where O(s) is the extremal odd integer of complexity at most s, determined in closed form by a three-branch expression governed by s mod 3. The powers-of-two conjecture follows as an immediate special case. All results are verified computationally for n <= 1000 and all complexity budgets K <= 20, with zero exceptions.